# Mahler takes a regular view of Zaremba

## TL;DR

This paper explores the properties of regular sequences associated with Zaremba's conjecture, analyzing their generating functions and Mahler equations to deepen understanding of continued fractions and their denominators.

## Contribution

It introduces a novel connection between Zaremba's conjecture and regular sequences, analyzing their generating functions and algebraic relations.

## Key findings

- Determined the minimal algebraic relation for the generating function.
- Established properties of the regular sequences related to Zaremba's conjecture.
- Analyzed Mahler iterates of the generating function.

## Abstract

In the theory of continued fractions, Zaremba's conjecture states that there is a positive integer $M$ such that each integer is the denominator of a convergent of an ordinary continued fraction with partial quotients bounded by $M$. In this paper, to each such $M$ we associate a regular sequence---in the sense of Allouche and Shallit---and establish various properties and results concerning the generating function of the regular sequence. In particular, we determine the minimal algebraic relation concerning the generating function and its Mahler iterates.